A377285 Position of first 0 in the n-th differences of the strict partition numbers A000009, or 0 if 0 does not appear.

0, 1, 1, 5, 5, 8, 20, 7, 22

Offset

0,4

Comments

Open problem: Do the 9th differences of the strict integer partition numbers contain a zero? If so, we must have a(9) > 10^5.

a(12) = 47. Conjecture: a(n) = 0 for n > 12. - Chai Wah Wu, Dec 15 2024

Links

Table of n, a(n) for n=0..8.

Example

The 7th differences of A000009 are: 25, -16, 7, -6, 10, -9, 0, 10, ... so a(7) = 7.

Mathematica

Table[Position[Differences[PartitionsQ/@Range[0, 100], k], 0][[1, 1]], {k, 1, 8}]

Prog

(PARI) a(n, nn=100) = my(q='q+O('q^nn), v=Vec(eta(q^2)/eta(q))); for (i=1, n, my(w=vector(#v-1, k, v[k+1]-v[k])); v = w; ); my(vz=select(x->x==0, v, 1)); if (#vz, vz[1]); \\ Michel Marcus, Dec 15 2024

Crossrefs

For primes we have A376678.

For composites we have A377037.

For squarefree numbers we have A377042.

For nonsquarefree numbers we have A377050.

For prime-powers we have A377055.

Position of first zero in each row of A378622. See also:

- A175804 is the version for partitions.

- A293467 gives first column (up to sign).

- A378970 gives row-sums.

- A378971 gives row-sums of absolute value.

A000009 counts strict integer partitions, differences A087897, A378972.

A000041 counts integer partitions, differences A002865, A053445.

Cf. A047966, A098859, A225486, A325244, A325282.

Cf. A008284, A116608, A325242, A325268, A225485 or A325280.

Sequence in context: A212533 A081287 A303715 * A204188 A347682 A334383

Adjacent sequences: A377281 A377282 A377283 * A377286 A377287 A377288

Keyword

nonn,more,new

Author

Gus Wiseman, Dec 12 2024

Status

approved